Question

College graduates are getting more for their degrees as starting salaries rise. To compare the starting salaries of college graduates majoring in chemical engineering and computer science, random samples of 50 recent college graduates in each major were selected and the following information obtained. $$\begin{array}{lll}\text { Major } & \text { Mean } & \text { SD } \\\\\hline \text { Chemical engineering } & \$ 53,659 & 2225 \\\\\text { Computer science } & 51,042 & 2375\end{array}$$ a. Find a point estimate for the difference in starting salaries of college students majoring in chemical engineering and computer science. What is the margin of error for your estimate? b. Based upon the results in part a, do you think that there is a significant difference in starting salaries for chemical engineers and computer scientists? Explain.

Short answer

Answer: To determine if there is a significant difference in starting salaries, we need to compare the point estimate and margin of error. If the margin of error is small compared to the point estimate and does not contain zero, we can infer that there is a significant difference in starting salaries.

Step-by-step solution

Calculate the point estimate for the difference in starting salaries

To find the point estimate for the difference in starting salaries, subtract the mean salary of computer science majors from the mean salary of chemical engineering majors. $$Point\,Estimate = Mean_{chemical} - Mean_{computer} = 53,659 - 51,042$$

Calculate the standard error

To find the standard error, follow this formula: $$SE = \sqrt{\frac{SD_{chemical}^2}{n_{chemical}}+\frac{SD_{computer}^2}{n_{computer}}}$$ where \(n_{chemical}\) is the sample size for chemical engineering majors and \(n_{computer}\) is the sample size for computer science majors. In this case, both sample sizes are 50. Using the given standard deviations, we get: $$SE = \sqrt{\frac{2225^2}{50}+\frac{2375^2}{50}}$$

Calculate the margin of error

To find the margin of error (ME), multiply the standard error by the appropriate critical value from the t-distribution, which would depend on the significance level and degrees of freedom. With a sample size of 50 in each group and no information regarding the significance level, let's use a common 95% confidence level and the corresponding critical value based on a two-tailed t-distribution with 49 (50-1) degrees of freedom. The critical value is approximately 2.01. $$ME = SE * critical\, value = SE * 2.01$$

Determine if there is a significant difference in starting salaries

To answer this question, we can look at the point estimate and margin of error. If the margin of error is large compared to the point estimate, then there is less certainty that there is a significant difference in starting salaries. If the margin of error is small compared to the point estimate, then it is more likely that there is a significant difference. Additionally, if the margin of error does not contain zero, we can strongly infer that there is a significant difference in starting salaries.

Key concepts

Point Estimate in Statistics

Understanding a point estimate in statistics is crucial for interpreting many types of data analysis, especially when dealing with averages or differences between groups. In the context of starting salaries for college graduates, we consider the mean salary for each major as a point estimate—an educated guess at a population parameter based on sample data. The mean salary for chemical engineering and computer science graduates, in this case, provide two separate point estimates.

The difference in these means, which we calculated as \( 53,659 - 51,042 \), serves as a point estimate for the difference in starting salaries between the two groups of graduates. This estimate is taken directly from sample means and gives us an initial comparison between the two majors in the real-world context, without remarking on the reliability or precision of this estimate, which is where the next concepts come into play.

Standard Error Calculation

The standard error (SE) is the measure of the amount of sampling variability in our estimate. It quantifies how spread out our point estimates would be if we took multiple samples from the same population. This stability—or lack thereof—helps us understand how much we can trust a single point estimate.

To calculate the SE, we use the formula provided in Step 2 of our solution. By plugging in the standard deviations (SDs) and sample sizes (n) for chemical and computer science majors, we obtained the SE for the difference in starting salaries. The formula incorporates both majors' variability, considering the sample sizes. Again, the larger the SE, the more spread out our estimates would be in repeated samples, and therefore, the less confident we can be in the accuracy of our point estimate.

Margin of Error

Once the standard error is determined, we can calculate the margin of error (ME). The ME provides a range, above and below the point estimate, creating an interval that likely contains the true difference in starting salaries if we were to look at all graduates. This interval is also known as the confidence interval.

In the example provided, after calculating the SE, we used a critical value from a t-distribution to find our ME. This critical value is associated with a desired confidence level, typically 95% in many social sciences—which informally means we are 95% confident that our interval contains the true mean difference. It's essential to note that a wider margin of error corresponds to a less precise estimate. If the margin does not include zero, as noted in Step 4, we could infer a significant difference. The ME is a crucial component of inferential statistics as it gives context and practical meaning to the point estimate.

Significance Level in Hypothesis Testing

The significance level, commonly denoted as alpha (\(\alpha\)), in hypothesis testing, is a threshold for deciding when to reject the null hypothesis. It is essentially the probability of making a Type I error—the error of rejecting a true null hypothesis. A common alpha level used in practice is 0.05, which equates to a 5% risk of concluding that a difference exists when there is actually none.

In the exercise, we analyzed the margin of error in the context of a 95% confidence level which indirectly relates to a significance level of 0.05. If we reject the null hypothesis that there is no difference in starting salaries, we are saying that the difference in our sample is significant enough that it is unlikely to have occurred just by random chance given our alpha level. In other words, it would be an uncommonly large difference if the true difference were actually zero.